De Sitter Momentum Space Achieves Novel , Dimensional Harmonic Expansion for Cosmology

Scientists are tackling the challenge of understanding field theory within the expanding universe, and a new study by Nathan Belrhali, Arthur Poisson, and Sébastien Renaux-Petel, all from the Institut d’Astrophysique de Paris, CNRS, Sorbonne Université, alongside Denis Werth (also affiliated with the Max Planck Institute for Physics and others), presents a novel momentum space specifically tailored for de Sitter spacetime. Their work constructs a non-perturbative momentum space , dubbed the Kontorovitch-Lebedev-Fourier (KLF) space , which dramatically simplifies calculations in early Universe cosmology by transforming complex time integrals into manageable frequency-space integrals. This innovative approach not only mirrors the structural elegance of Minkowski space, offering algebraic equations of motion and a flat-space-like propagator, but also provides a natural framework for incorporating crucial spectral contributions, potentially revolutionising our ability to model quantum fields in a cosmological context.

Furthermore, the research establishes an inversion formula for determining the spectral density, offering a powerful tool for analysing quantum fields in dS spacetime. This momentum space provides a nonperturbative definition of quantum field theory, free from infrared divergences, and uniquely defines the Bunch-Davies vacuum. Experiments show that the KLF space, dual to Poincaré coordinates (τ, x) and labelled by (μ, k), offers a transparent view of dS symmetries, dynamics, and analytic structure.

Equations of motion reduce to algebraic relations, and the Feynman propagator assumes a form identical to that in flat space, significantly simplifying calculations. The team achieved this by recognizing that conventional Fourier transforms fail to diagonalize dS isometries, leading to entangled symmetry constraints, dynamics, and analytic structures, issues resolved by their new representation. This approach streamlines computations and provides a clear physical interpretation of complex calculations. Researchers proved that the resulting integrands are universally meromorphic, allowing for the evaluation of correlators by simply collecting residues, a significant computational advantage. The work opens new avenues for exploring the analytic structure of late-time correlators, which have historically been difficult to compute, and for understanding the loop-level behaviour of cosmological observables. By considering (d+1)-dimensional dS spacetime embedded in (d+2)-dimensional Minkowski spacetime, defined by ηABXAXB = H−2, the study provides a general framework adaptable to dimensional regularization and future cosmological investigations.

Kontorovitch-Lebedev-Fourier Space on de Sitter Spacetime

This KLF space allows equations of motion to reduce to algebraic equations, and the quadratic dynamics yields a propagator remarkably analogous to flat space, simplifying complex calculations. The study meticulously defined the experimental setup by considering (d+1)-dimensional dS spacetime embedded within (d+2)-dimensional Minkowski spacetime, utilising coordinates where X0 = (2Hτ)−1(τ 2−x2− 1), Xi = −(Hτ)−1xi, and Xd+1 = (2Hτ)−1(x2 −τ 2 −1), with conformal time τ ranging from −∞ to 0. The metric induced on the hyperboloid, ds2 = (Hτ)−2 −dτ 2 + dx2, accurately describes an accelerating spacetime with a constant Hubble rate H and a scale factor of −(Hτ)−1, providing a precise cosmological framework. Researchers then decomposed the Hilbert space of QFT in dS into Unitary Irreducible Representations (UIRs) of SO(1, d + 1), labelled by μ, where each UIR corresponds to an eigenvalue M2(μ) of the quadratic Casimir operator C = −1 2JABJAB.

Experiments employed a classification of UIRs based on eigenvalues, revealing three categories: principal series P(μ) with μ ∈ R, complementary series C(μ) with iμ ∈ [−d 2, d 2], and type-I exceptional series E(μ) with iμ = d 2 + k, where k is a non-negative integer. The team parameterised the Casimir eigenvalues as M2(μ) = μ2 + d2 4, highlighting a Z2 shadow symmetry μ ↔ −μ, and restricted their attention to scalar representations with S = 0, simplifying the analysis. Furthermore, the system delivers an inversion formula for the spectral density, enabling the evaluation of correlators by simply collecting residues, a significant advancement in computational efficiency! The work details how this KLF momentum space simultaneously clarifies dS symmetries, dynamics, and analyticity, offering a uniquely defined, nonperturbative definition of QFT free from infrared divergences and a uniquely defined (Euclidean) Bunch-Davies vacuum, ultimately providing a powerful tool for exploring the complexities of early Universe cosmology.

Kontorovitch-Lebedev-Fourier Space on de Sitter Spacetime

This KLF space exhibits structural similarities to its Minkowski counterpart, notably reducing equations of motion to algebraic forms and yielding a flat-space-like propagator for quadratic dynamics. Experiments revealed that the distribution δα(μ) acts in KLF space as an integral, resulting in a symmetrization that enforces shadow symmetry; for real α, this simplifies to δα(μ) = 4 |α|δ(μ2 −α2). This key aspect of the formalism allows computations to be initially performed for fields in the principal series, circumventing issues like infrared divergences, with more general physical results obtained through analytical continuation. Researchers defined KLF-space correlators for real μi as Gan. an μ1. μn k1. kn, establishing a functional generator Z[J+, J−] integrated over L2 EAdSd+1 functions.

Measurements confirm that the action integrals are taken over two in-in branches with integration axes tilted in the complex time plane, sewn together at a late time τ0 where φ+(τ0) = φ−(τ0). For a Gaussian action, the analytically continued form in EAdS, decomposed onto the KLF harmonic basis, yields ±iS0,±[φ] = H2e± i(d−1)π 2 2 Z KLF φ(μ) k (μ2 −μ2 φ)φ(μ) −k. The breakthrough delivers four KLF-space propagators, defined by G′ μφ(μ) = e−iπ(d+1) 2 (μ2−μ2φ)iε δμφ(μ) Nμ δμφ(μ) Nμ e+ iπ(d+1) 2 (μ2−μ2 φ)−iε. Tests prove that the iε prescription, encoding particle production, is given by 1/(μ2 −μ2φ)iε ≡ 1/2 [sinh(πμφ) e+πμφ / (μ2 −μ2φ + iε) − e−πμφ / (μ2 −μ2φ − iε)].

The diagonal terms in the propagator matrix correspond to Minkowski Feynman propagators in momentum space, G′(p) = ±i/(−p2 −m2 φ ± iε). Scientists formulated Feynman rules for KLF correlators, where each vertex carries a SK ± index, propagators are defined by equation (21), and polynomial interactions contribute terms proportional to −iaλH (n−2)(d+1) 2 ea iπd 2 π n 2 Iμ1. μn k1. kn. Data shows that amputated KLF correlators link KLF space to observable quantities, with physical correlators obtained by summing diagrams over ± contributions, ensuring opposite SK indices for external legs and attached vertices. Researchers analytically continued KLF correlators to arbitrary complex μi, demonstrating that the s-channel amputated KLF diagram arising from −1 2λφ2χ is proportional to (−iλ)2 Hd+1e−iπd 2 π3 +∞ Z −∞ dμNμ Iμ1μ2μ k1k2s Iμμ3μ4 sk3k4 (μ2 −μ2χ)iε. The spectral decomposition of correlators was derived from first principles, based on symmetry, revealing that the Wightman two-point function can be expressed as an integral over the spectral density ρP,C O (μ)Φ(μ) k.

Kontorovitch-Lebedev-Fourier Space and dS QFT Reformulation

This framework allows for a direct reformulation of QFT, conveniently defined on a specific path integral contour, and establishes a clear correspondence between cosmological and amputated correlators. Key features made manifest by this approach include the nonconservation of dS frequency, the natural emergence of Bessel-type dS harmonic functions for time/frequency duality, and a restructuring of perturbative calculations where time ordering is replaced by spectral integration. Furthermore, the KLF transform directly isolates the spectral density of local operators at the nonperturbative level, offering a transparent view of the underlying structure of cosmological correlators and facilitating connections to flat space physics. The authors acknowledge limitations in the current work, specifically noting that the framework is presently limited to the Poincaré patch of dS spacetime. Future research will explore the potential generalization or deformation of KLF space to accommodate symmetry breaking of dS isometries, and investigate the use of this tool for propagator dressing and correlator resummation. Connecting to flat-space amplitudes, where energy conservation arises from spectral integration, and defining an S-matrix from KLF amputated correlators are also avenues for future investigation, promising fundamental new insights into QFT in dS spacetime.